Course detail

Mathematical Methods in Fluid Dynamics

FSI-SMMCompulsoryMaster's (2nd cycle)Acad. year: 2016/2017Winter semester2. year of study4  credits

Basic physical laws of continuum mechanics: laws of conservation of mass, momentum and energy. Theoretical study of hyperbolic conservation laws, particularly of Euler equations that describe the motion of inviscid compressible fluids. Numerical modelling based on the finite volume method. Numerical modelling of incompressible flows: Navier-Stokes equations, pressure-correction method, spectral element method.

Learning outcomes of the course unit

Students will be made familiar with basic principles of the fluid flow modelling: physical laws, the mathematical analysis of equations describing flows (Euler and Navier-Stokes equations), the choice of an appropriate method (which issues from the physical as well as from the mathematical essence of equations) and the computer implementation of proposed method (preprocessing = mesh generation, numerical solver, postprocessing = visualization of desired physical quantities). Students will demonstrate the acquinted knowledge by elaborating semester assignement.

Mode of delivery

90 % face-to-face, 10 % distance learning


Evolution partial differential equations, functional analysis, numerical methods for partial differential equations.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

M. Feistauer, J. Felcman, I. Straškraba: Mathematical and Computational Methods for Compressible Flow, Oxford University Press, Oxford, 2003
L. Čermák: Výpočtové metody dynamiky tekutin, dostupné na
V. Dolejší, M. Feistauer: Discontinuous Galerkin Method, Springer, Heidelberg, 2016.
E.F. Toro: Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, Springer, Berlin, 1999.
J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics, Springer-Verlag, New York, 2002.
K. H. Versteeg, W. Malalasekera: An Introduction to Computational Fluid Dynamics, Pearson Prentice Hall, Harlow, 2007.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

COURSE-UNIT CREDIT IS AWARDED ON THE FOLLOWING CONDITIONS: Active participation in practicals. Elaboration of a semester assignment, where the students prove their knowledge acquired. Students, who gain course-unit credits, will also obtain 0--30 points, which will be included in the final course classification.
FORM OF EXAMINATIONS: The exam is oral. As a result of the exam students will obtain 0--70 points.
FINAL ASSESSMENT: The final point course classicifation is the sum of points obtained from both the practisals (0--30) and the exam (0--70).
FINAL COURSE CLASSIFICATION: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.
If we measure the exam success in percentage points, then the classification grades are: A (excellent): 100--90, B (very good): 89--80, C (good): 79--70, D (satisfactory): 69--60, E (sufficient): 59--50, F (failed): 49--0.

Language of instruction


Work placements

Not applicable.


The course is intended as an introduction to the computational fluid dynamics. Considerable emphasis will be placed on the inviscid compressible flow: namely, the derivation of Euler equations, properties of hyperbolic systems and an introduction of several methods based on the finite volumes. Methods for computations of viscous flows will be also studied, namely the pressure-correction method and the spectral element method. Students ought to realize that only the knowledge of substantial physical and mathematical aspects of particular types of flows enables them to choose an effective numerical method and an appropriate software product. The development of individual semester assignement constitutes an important experience enabling to verify how the subject matter was managed.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is required. Lessons are planned according to the week schedules. Absence from lessons may be compensated by the agreement with the teacher supervising the seminars.

Type of course unit



26 hours, optionally

Teacher / Lecturer


1. Material derivative, transport theorem, mass, momentum and energy conservation laws.
2. Constitutive relations, thermodynamic state equations, Navier-Stokes and Euler equations, initial and boundary conditions.
3. Traffic flow equation, acoustic equations, shallow water equations.
4. Hyperbolic system, classical and week solution, discontinuities.
5. The Riemann problem in linear and nonlinear case, wave types.
6. Finite volume method in one and two dimensions, numerical flux.
7. Local error, stability, convergence.
8. The Godunov's method, flux vector splitting methods: the Vijayasundaram, the Steger-Warming, the Van Leer.
9. Viscous incompressible flow: finite volume method for orthogonal staggered grids, pressure correction method SIMPLE.
10. Pressure correction method for colocated variable arrangements, non-orthogonal and unstructured meshes.
11. Stokes problem, spectral element method.
12. Steady Navier-Stokes problem, spectral element method.
13. Unsteady Navier-Stokes problem, time dicretization methods.

seminars in computer labs

13 hours, compulsory

Teacher / Lecturer


Demonstration of solutions of selected model tasks on computers. Elaboration of the semester assignment.